What is the best Linux disto for an old laptop with limited HDD space?

Advise please. I have an old Compaq Presario v5000 with 512 Mb RAM and a 40Gb HHD and 1.4gigahertz processor. It's about 9 or 10 years old. I want to completely replace the XP OS on it with a Linux distro. I tried the most recent Ubuntu, but it needed more hard disk space than I've got. What is the best Linux disto for an old laptop with limited HDD space? Also, I am disabled and have like zero bucks, so it needs to be a free distro. One last thing, the monitor on the laptop is broken so I replaced it with an old external monitor and I will need to be able to configure the Linux OS to fit it. Thanks.

I'm also a broke disabled person using an old laptop with a dead screen; mine...

I'm also a broke disabled person using an old laptop with a dead screen; mine is a 1.6GHz that I upgraded to 1 GB of memory with a $5 stick from eBay a couple of years ago.

What really matters isn't the distro itself as much as the desktop environment. (99% of distros are completely free, so that's not an issue.) The easiest approach would be to download one of the live CDs that are hosted directly by a lightweight desktop team -- the two I know of that have some of their own CDs to download & try out are LXDE and Trinity.

When you've gotten used to whichever one you end up using, and feel like you want to try something new, you'll almost certainly know enough about Linux to figure out what to try next. Good luck!